Firstly you definitely have time to package your products. For carrier service - (products that need to be shipped) - for ground shipping, we work with UPS using UPS Smart Pickup. Once you have signed the agreement with UPS, you will be added to the UPS local route and they will stop by 3x week on a regularly scheduled basis. That way there will be no surprises.
For orders that are overnight-ship or 2 day, please package those immediately as they will be picked up immediately. Your customers are paying much higher rates to get that product ASAP.
If you really can't find the time, there are on-demand services such as TaskRabbit where you put in what you need to be done and someone comes and does it. Ex. I need to box up 30x 6 packs. I estimate it will take 2 hours. Then you receive hourly or project bids from a number of local people in your area.
Regarding local, on-demand delivery, (we pick up the product as if we are a normal customer at the counter) based on your open hours of operation. This allows our drivers to show up with a receipt and get the product. I.e. growler to-go, 6 pack to-go, etc. No packing required!